chapter18presentation.ppt

1、1,Chapter 18: The Chi-Square Statistic,2,Parametric and Nonparametric Tests,Chapter 18 introduces two non-parametric hypothesis tests using the chi-square statistic: the chi-square test for goodness of fit and the chi-square test for independence.,3,Parametric and Nonparametric Tests (cont.),The ter

2、m non-parametric refers to the fact that the chisquare tests do not require assumptions about population parameters nor do they test hypotheses about population parameters. Previous examples of hypothesis tests, such as the t tests and analysis of variance, are parametric tests and they do include a

3、ssumptions about parameters and hypotheses about parameters.,4,Parametric and Nonparametric Tests (cont.),The most obvious difference between the chisquare tests and the other hypothesis tests we have considered (t and ANOVA) is the nature of the data. For chisquare, the data are frequencies rather

4、than numerical scores.,5,The Chi-Square Test for Goodness-of-Fit,The chi-square test for goodness-of-fit uses frequency data from a sample to test hypotheses about the shape or proportions of a population. Each individual in the sample is classified into one category on the scale of measurement. The

5、 data, called observed frequencies, simply count how many individuals from the sample are in each category.,6,The Chi-Square Test for Goodness-of-Fit (cont.),The null hypothesis specifies the proportion of the population that should be in each category. The proportions from the null hypothesis are u

6、sed to compute expected frequencies that describe how the sample would appear if it were in perfect agreement with the null hypothesis.,8,The Chi-Square Test for Independence,The second chi-square test, the chi-square test for independence, can be used and interpreted in two different ways: 1. Testi

7、ng hypotheses about the relationship between two variables in a population, or 2. Testing hypotheses about differences between proportions for two or more populations.,9,The Chi-Square Test for Independence (cont.),Although the two versions of the test for independence appear to be different, they a

8、re equivalent and they are interchangeable. The first version of the test emphasizes the relationship between chi-square and a correlation, because both procedures examine the relationship between two variables.,10,The Chi-Square Test for Independence (cont.),The second version of the test emphasize

9、s the relationship between chi-square and an independent-measures t test (or ANOVA) because both tests use data from two (or more) samples to test hypotheses about the difference between two (or more) populations.,11,The Chi-Square Test for Independence (cont.),The first version of the chi-square te

10、st for independence views the data as one sample in which each individual is classified on two different variables. The data are usually presented in a matrix with the categories for one variable defining the rows and the categories of the second variable defining the columns.,12,The Chi-Square Test

11、 for Independence (cont.),The data, called observed frequencies, simply show how many individuals from the sample are in each cell of the matrix. The null hypothesis for this test states that there is no relationship between the two variables; that is, the two variables are independent.,13,The Chi-S

12、quare Test for Independence (cont.),The second version of the test for independence views the data as two (or more) separate samples representing the different populations being compared. The same variable is measured for each sample by classifying individual subjects into categories of the variable

13、. The data are presented in a matrix with the different samples defining the rows and the categories of the variable defining the columns.,14,The Chi-Square Test for Independence (cont.),The data, again called observed frequencies, show how many individuals are in each cell of the matrix. The null h

14、ypothesis for this test states that the proportions (the distribution across categories) are the same for all of the populations,15,The Chi-Square Test for Independence (cont.),Both chi-square tests use the same statistic. The calculation of the chi-square statistic requires two steps: 1.The null hy

15、pothesis is used to construct an idealized sample distribution of expected frequencies that describes how the sample would look if the data were in perfect agreement with the null hypothesis.,16,The Chi-Square Test for Independence (cont.),For the goodness of fit test, the expected frequency for eac

16、h category is obtained by expected frequency = fe = pn (p is the proportion from the null hypothesis and n is the size of the sample) For the test for independence, the expected frequency for each cell in the matrix is obtained by (row total)(column total) expected frequency = fe = n,18,The Chi-Squa

17、re Test for Independence (cont.),2. A chi-square statistic is computed to measure the amount of discrepancy between the ideal sample (expected frequencies from H0) and the actual sample data (the observed frequencies = fo). A large discrepancy results in a large value for chi-square and indicates th

18、at the data do not fit the null hypothesis and the hypothesis should be rejected.,19,The Chi-Square Test for Independence (cont.),The calculation of chi-square is the same for all chi-square tests: (fo fe)2 chi-square = 2 = fe The fact that chisquare tests do not require scores from an interval or r

19、atio scale makes these tests a valuable alternative to the t tests, ANOVA, or correlation, because they can be used with data measured on a nominal or an ordinal scale.,20,Measuring Effect Size for the Chi-Square Test for Independence,When both variables in the chi-square test for independence consist of exactly two categories (t